USEFUL FORMULAE IN DIFFERENTIAL GEOMETRY
Differential forms:
1 α = α dxµ1 ∧ · · · ∧ dxµp ; α ∈ ∧p. (1) p! µ1...µp
α ∧ β = (−)pqβ ∧ α; α ∈ ∧p, β ∈ ∧q. (2)
Exterior derivative, d:
1 dα ≡ ∂ α dxv ∧ dxµ1 ∧ · · · ∧ dxµp . (3) p! [ν µ1...µp ] d maps p-forms to (p + 1)-forms:
d : ∧p → ∧p+1; d2 = 0. (4)
Defining the components of dα,(dα)µ1...µp+1 , by
1 dα ≡ (dα) dxµ1 ∧ · · · ∧ dxµp+1 , (5) (p + 1)! µ1...µp+1 we have
(dα)µ1...µp+1 = (p + 1)∂[µ1 αµ2...µp+1], (6) where 1 T ≡ T + even perms − odd perms . (7) [µ1...µq] q! µ1...µq Leibnitz rule: d(α ∧ β) = dα ∧ β + (−)pα ∧ dβ, α ∈ ∧p, β ∈ ∧q. (8)
Stokes’ Theorem: Z Z dω = ω, (9) M ∂M where M is an n-manifold and ω ∈ ∧n−1.
Epsilon tensors and densities:
εµ1...µn ≡ (+1, −1, 0) (10) if µ1 . . . µn is an (even, odd, no) permutation of a lexical ordering of indices (1 . . . n). It is a tensor density of weight +1. We may also define the quantity εµ1···µn , with components given numerically by µ1···µn t ε ≡ (−1) εµ1···µn ,
1 where t is the number of timelike coordinates. NOTE: This is the only quantity where we do not raise and lower indices using the metric tensor. εµ1...µn is a tensor density of weight −1. We define epsilon tensors: 1 p µ1...µn µ1...µn µ ...µ = |g| εµ ...µ , = ε , (11) 1 n 1 n p|g|
µ ...µn where g ≡ det(gµν) is the determinant of the metric tensor gµν. Note that the tensor 1
is obtained from µ1...µn by raising the indices using inverse metrics.
Epsilon-tensor identities: ν1...νn t ν1...νn µ1...µn = (−1) n! δµ1...µn . (12a) From this, contractions of indices lead to the special cases
µ1...µrνr+1...νn t νr+1...νn µ1...µrµr+1...µn = (−1) r!(n − r)! δµr+1...µn , (12b) where again t denotes the number of timelike coordinates. The multi-index delta-functions have unit strength, and are defined by
ν1···νp [ν1 νp] δµ ···µ ≡ δ ··· δ . (13) 1 p [µ1 µp] (Note that only one set of square brackets is actually needed here; but with our “unit- strength” normalisation convention (7), the second antisymmetrisation is harmless.) It is worth pointing out that a common occurrence of the multi-ndex delta-function is in an ν1···νp expression like Bν1 Aν2···νp δµ1···µp , where Aν2···νp is totally antisymmetric in its (p−1) indices. It is easy to see that this can be written out as the p terms 1 B A δν1···νp = B A + B A + B A + ··· + B A ν1 ν2···νp µ1···µp p µ1 µ2···µp µ2 µ3···µpµ1 µ3 µ4···µpµ1µ2 µp µ1···µp−1 if p is odd. If instead p is even, the signs alternate and 1 B A δν1···νp = B A −B A +B A −· · ·−B A . ν1 ν2···νp µ1···µp p µ1 µ2···µp µ2 µ3···µpµ1 µ3 µ4···µpµ1µ2 µp µ1···µp−1
Hodge ∗ operator:
1 ∗(dxµ1 ∧ · · · ∧ dxµp ) ≡ µ1...µp dxν1 ∧ · · · ∧ dxνn−p . (14) (n − p)! ν1...νn−p The Hodge ∗, or dual, is thus a map from p-forms to (n − p)-forms:
∗ : ∧p → ∧n−p. (15)
Note in particular that taking p = 0 in (14) gives
1 p ∗1 = = dxµ1 ∧ · · · ∧ dxµn = |g| dx1 ∧ · · · ∧ dxn. (16) n! µ1...µn 2 This is the general-coordinate-invariant volume element p|g| dnx of Riemannian geometry. It should be emphasised that conversely, we have p dxµ1 ∧ dxµ2 ∧ · · · ∧ dxµn = (−1)t εµ1µ2···µn dnx = (−1)t µ1µ2···µn |g| dnx .
This extra (−1)t factor is tiresome, but unavoidable if we want our definitions to be such that ∗1 is always the positive volume element. From these definitions it follows that 1 ∗α ∧ β = |α · β| , (17) p! where α and β are p-forms and
µ1...µp |α · β| ≡ αµ1...µp β . (18)
Applying ∗ twice, we get
∗ ∗ ω = (−)p(n−p)+t ω, ω ∈ ∧p. (19)
In even dimensions, n = 2m, m-forms can be eigenstates of ∗, and hence can be self- dual or anti-self-dual, in cases where ∗∗ = +1. From (19), we see that this occurs when m is even if t is even, and when m is odd if t is odd. In particular, we can have real self- duality and anti-self-duality in n = 4k Euclidean-signature dimensions, and in n = 4k + 2 Lorentzian-signature dimensions.
Adjoint operator, δ: First define the inner product
Z 1 Z (α, β) ≡ ∗α ∧ β = |α · β| = (β, α), (20) M p! M where α and β are p-forms. Then δ, the adjoint of the exterior derivative d, is defined by
(α, dβ) ≡ (δα, β), (21)
where α is an arbitrary p-form and β is an arbitrary (p − 1)-form. Hence
δα = (−)np+t∗d∗α, α ∈ ∧p. (22)
(We assume that the boundary term arising from the integration by parts gives zero, either because M has no boundary, or because appropriate fall-off conditions are imposed on the fields.) δ is a map from p-forms to (p − 1)-forms:
δ : ∧p → ∧p−1; δ2 = 0. (23)
3 Note that in Euclidean signature spaces, δ on p-forms is given by
δα = ∗d∗α if at least one of n and p even, (24) δα = −∗d∗α, if n and p both odd.
The signs are reversed in Lorentzian spacetimes. In terms of components, the above definitions imply that for all spacetime signatures, we have 1 δα = − (∇ αν ) dxµ1 ∧ · · · ∧ dxµp−1 , (25) (p − 1)! ν µ1...µp−1 where 1 √ ∇ ανµ1...µp−1 ≡ √ ∂ gανµ1...µp−1 (26) ν g ν is the covariant divergence of α. Defining the components of δα,(δα)µ1...µp−1 , by 1 δα ≡ (δα) dxµ1 ∧ · · · ∧ dxµp−1 , (27) (p − 1)! µ1...µp−1
we have ν (δα)µ1...µp−1 = −∇να µ1...µp−1 . (28)
Hodge-de Rham operator:
∆ ≡ dδ + δd = (d + δ)2. (29) ∆ maps p-forms to p-forms: ∆ : ∧p → ∧p. (30) On 0-, 1-, and 2-forms, we have
λ 0-forms: ∆φ = −∇λ∇ φ, λ ν 1-forms: ∆ωµ = −∇λ∇ ωµ + Rµ ων, (31) λ ρσ σ σ 2-forms: ∆ωµν = −∇λ∇ ωµν − 2Rµρνσω + Rµ ωσν − Rν ωσµ,
where Rµνρσ is the Riemann tensor and
ρ Rµν ≡ R µρν (32)
is the Ricci tensor. Hodge’s theorem: We can uniquely decompose an arbitrary p form ω as
ω = dα + δβ + ωH , (33)
p−1 p+1 where α ∈ ∧ , β ∈ ∧ and ωH is harmonic, ∆ωH = 0.
4 RIEMANNIAN GEOMETRY
2 µ ν a For a metric ds = gµνdx dx , we define a vielbein eµ as a “square root” of gµν:
a b gµν = eµ eν ηab, (34)
where ηab is a local Lorentz metric. Usually, we work with positive-definite metric signature, µ so ηab = δab. The inverse vielbein, which we denote by Ea , satisfies
µ a µ µ b b Ea eν = δν ; Ea eµ = δa. (35)
a a µ The “solder forms” e = eµ dx give an orthonormal basis for the cotangent space. Similarly, µ the vector fields Ea ∂µ give an orthonormal basis for the tangent space.
Torsion and curvature
a a µ a We define the spin connection ω b = ωµbdx , the torsion 2-form T and the curvature a 2-form Θ b by
a 1 a µ ν a a b T ≡ 2 Tµνdx ∧ dx = de + ω b ∧ e , (36) a 1 a µ ν a a c Θ b ≡ 2 R bµνdx ∧ dx = dω b + ω c ∧ ω b. (37)
Define a Lorentz-covariant and general-coordinate covariant derivative Dµ that acts on tensors with coordinate and Lorentz indices:
νa νa a νc c νa Dµ Vρb = ∇µ Vρb + ωµcVρb − ωµbVρc , (38)
where ∇µ is the usual general-coordinate covariant derivative:
ν ν ν σ σ ν ∇µ Vρ = ∂µ Vρ + ΓµσVρ − ΓµρVσ , (39)
µ and Γνρ is the Christoffel connection. Demanding metricity for gµν, i.e. Dµ gνρ = 0, implies
µ 1 µσ Γνρ = 2 g ∂ν gσρ + ∂ρ gνσ − ∂σ gνρ . (40)
Demanding metricity for ηab, i.e. Dµ ηab = 0, implies
ωab = −ωba, (41)
c where ωab ≡ ηacω b. Bianchi Identities Taking exterior derivative of (36) and (37) gives
a a a b a b DT ≡ d T + ω b ∧ T = Θ b ∧ e , (42) a a a c a c D Θ b ≡ d Θ b + ω c ∧ Θ b − Θ c ∧ ω b = 0. (43)
5 a In general, on Lorentz-valued p forms such as α b, we define the Lorentz-covariant exterior derivative by a a a c c a D α b ≡ d α b + ω c ∧ α b − ω b ∧ α c. (44)
Torsion-free metric connection With the metricity assumption, implying (41), and the assumption that the torsion a vanishes, it follows that ω b is then uniquely determined by (36) and (41);
a a b d e = −ω b ∧ e ; ωab = −ωba. (45)
c c Defining cab = −cba by a 1 a b c d e = − 2 cbc e ∧ e , (46) it follows that ωab is given by
1 c ωab = 2 (cabc + cacb − cbca)e . (47) Note that the vielbein is constant with respect to the Lorentz- and general-coordinate co- a variant derivative defined by (38); Dµ eν = 0. Symmetries of the Riemann tensor It follows from its definition as 2-form (37) that it is always antisymmetric on the final index pair: Rabµν = −Rabνµ; Rabcd = −Rabdc, (48) where we can always freely convert coordinates indices to Lorentz indices, and vice versa, µ ν c d using the vielbein. Thus Rabcd = Ec Ed Rabµν and conversely Rabµν = eµeνRabcd. The metricity condition Dµηab = 0 implies ωab = −ωba, and hence Θab = −Θba. Thus
Rabcd = −Rbacd. Metricity (49)
The torsion-free condition, using (42), implies that
Ra[bcd] = 0, Torsion-free (50)
1 where Ra[bcd] = 3 (Rabcd + Racdb + Radbc). Together, (48), (49) and (50) imply
Rabcd = Rcdab. (51)
The Ricci tensor and scalar, and Weyl tensor
We define the Ricci tensor Rab and Ricci scalar R by
c ab Rab ≡ R acb; R ≡ Rabη . (52)
Note that (51) implies that the Ricci tensor is symmetric, Rab = Rba.
6 The Weyl tensor Cabcd is defined in n dimensions by 1 C ≡ R − R η − R η + R η − R η abcd abcd n − 2 ac bd ad bc bd ac bc ad 1 (53) + R η η − η η . (n − 1)(n − 2) ac bd ad bc
c It is the “traceless” part of the Riemann tensor, in the sense that C acb ≡ 0. It has the same symmetries (48)-(51) as the Riemann tensor for torsion-free connection. One may define the Weyl 2-form Ωab,
1 c d Ωab ≡ 2 Cabcde ∧ e 1 1 (54) = Θ − R η − R η ec ∧ ed + Rη η ec ∧ ed. ab n − 2 ac bd bc ad (n − 1)(n − 2) ac bd
YANG-MILLS THEORY
If ϕ is a set of scalar fields in some representation R of a Lie group G, then we define ϕ0, the gauge-transformed field, by ϕ0 = h−1ϕ, (55) where h = h(x) is a map from the base space M into the group G. The Yang-Mills covariant derivative D of ϕ is defined to be
Dϕ ≡ (d + A)ϕ, (56) where the Yang-Mills potential, or connection, A, taking its values in the adjoint represen- tation of G, is defined to transform under gauge transformations as
A0 ≡ h−1Ah + h−1dh. (57)
It then follows that Dϕ indeed transforms in the desired covariant manner, namely
(Dϕ)0 ≡ D0ϕ0 = h−1Dϕ. (58)
The Yang-Mills field strength, or curvature, F , is defined by
F ≡ dA + A ∧ A. (59)
Under gauge transformations, it transforms covariantly, as
F 0 = h−1F h. (60)
The infinitesimal forms of these transformations, when h = 1 + Λ, where Λ is infinitesimal, reduce to the results derived in the lectures.
7 Kaluza-Klein and O’Neill’s formula
Given a base space M with metric ds2, and a principal bundle with fibre group G defined over it, with connection (Yang-Mills potential) A, we may write down a 1-parameter family of natural metrics on the bundle as
2 2 i 2 2 d˜s = λ (Σi − A ) + ds , (61) where λ is an arbitrary constant, and a summation over i = 1,..., dim(G) is understood. The Σi are left-invariant 1-forms on the group G, which means that they satisfy
1 dΣi = − 2 fijkΣj ∧ Σk, (62) where fijk = f[ijk] are the structure constants of the group. Then the Riemann tensor for the metric d˜s2 is given by
1 2 i i i i i i Reαβγδ = Rαβγδ − 4 λ FαγFβδ − FαδFβγ + 2FαβFγδ , 1 i Reαβγi = 2 λDγFαβ, 1 2 i j 1 k (63) Reαiβj = 4 λ FβγFαγ − 4 fijkFαβ, 1 Reijk` = fijmfk`m, 4λ2 together with those components related to the above by the Riemann tensor symmetries (48)-(51). Here we are taking the orthonormal basis
˜ei = λ(Σ − Ai), (i = 1,..., dim(G)), i (64) ˜eα = eα, (α = 1, . . . , n),
α 2 α α where e is an orthonormal basis for the base space M: thus ds = e e . Rαβγδ are the orthonormal components of the Riemann tensor on M, and
i i 1 j k F = dA + 2 fijkA ∧ A , i i j k DγFαβ = DγFαβ + fijkAγFαβ, (65) D F i = Eµ ∂ F i + ωαγF i + ωβγF i . γ αβ γ µ αβ µ γβ µ αγ
(So Dµ is the Lorentz-covariant derivative, and Dµ is the Lorentz and Yang-Mills covariant derivative.)
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